Charged-particle-beam pattern-transfer apparatus use a charged-particle beam, such as an electron beam, to project a pattern from a reticle or mask onto a wafer. In such apparatus, a rotationally-symmetric magnetic lens typically images the reticle onto the wafer. This lens must produce clear, undistorted images. One satisfactory lens type is a magnetic doublet having two magnetic lenses that produce opposite magnetic fields.
The trajectory of a charged particle in a charged-particle optical system is usually described in terms of a paraxial trajectory and aberrations. In a rotationally symmetric electric or magnetic field, the trajectory of a charged particle is calculated based on the coordinates (x.sub.0, y.sub.0) and propagation angles (.alpha..sub.1, .alpha..sub.2) of the particle at the wafer plane. The aberrations of the optical system are expressed in terms of polynomials of order 2n+1, where n.gtoreq.1. Terms of order 2n+1 represent the (2n+1)th order aberrations.
While most aberrations are represented by terms of order 3 or higher, chromatic aberration effects even the paraxial trajectory. Chromatic aberration is caused by the spread of charged particle energies in the charged particle beam and chromatic aberration therefore occurs at orders of (2n-1) or larger, where n.gtoreq.1.
If the charged particle trajectory is expressed in terms of the complex coordinates w=x+iy, the total third-order geometric aberration as a function of the complex coordinate .beta.=x.sub.0 +iy.sub.0 and the complex propagation angle .alpha.=.alpha..sub.1 +i.alpha..sub.2 at the wafer is given by: ##EQU2## wherein .alpha.* and .beta.* are the complex conjugates of .alpha. and .beta. respectively, V is the accelerating voltage and .DELTA.V is the charged-particle-beam energy spread. The various aberration coefficients are K.sub.sph (spherical aberration), K.sub.coma-l (longitudinal coma), K.sub.coma-r (transverse or radial coma), K.sub.fc (field curvature), K.sub.astig (astigmatism), K.sub.dis (distortion), K.sub.t-chro (transverse chromatic aberration), and K.sub.a-chro (axial chromatic aberration).
In a magnetic doublet, distortion, chromatic aberration, and image rotation generated by a first lens offset distortion, chromatic aberration, and image rotation produced by a second lens. For this reason, the transverse chromatic aberration coefficient K.sub.t-chro and the distortion coefficient K.sub.dis are zero. However, the magnetic doublet exhibits other types of aberrations that depend upon fields produced by the lens.
In charged-particle-beam pattern-transfer apparatus, image blur caused by Coulomb interactions of the charged particles in the beam limits throughput. In order to reduce this image blur, the beam current density is reduced. One method of decreasing current density without decreasing throughput is to irradiate a large area of the reticle with a large diameter beam. Increasing the beam numerical aperture also reduces image blur due to Coulomb interactions.
The aberrations of the magnetic doublet are not determined solely by the aberrations of the individual lenses of the doublet, and optimization of the individual lenses does not ensure optimization of the doublet. In addition, the actual aberrations realized during use depend upon the propagation direction of the beam and location of the beam on the reticle. For example, if the reticle is demagnified with a demagnification 1/m onto the wafer, then the initial coordinate of the beam is m.beta. at the reticle. If the initial coordinate of the beam is increased (i.e., the beam propagates farther off-axis), then the aberrations are changed according to Equation 1. For this reason, lens-system design calculations are complex and it is difficult to optimize such lenses.
Therefore, it is an object of the invention to reduce the blur associated with geometric aberrations and Coulomb interactions.